Sorry, this one is a little long-winded, but I hope it's clear! I've come across a problem that I'm struggling to solve because it *requires* the use of certain standard results as the starting point. The problem is to find the solution of

$\displaystyle \sum\limits_{r = 1}^n {\frac{1}{r\left( {r + 1} \right)}} $

I can solve this problem quite easily by considering the form of each term in the sum, or through the use of partial fractions (where I find an ultimate answer of $\displaystyle \frac{n}{n + 1}$, but I'm supposed to solve it using one (or more) of the following standard results (I'll include all of them, but I imagine only the final one is relevant):

$\displaystyle \sum\limits_{r = 1}^n {r} = \frac{1}{2} {n ( {n + 1} )} $

$\displaystyle \sum\limits_{r = 1}^n {r^2} = \frac{1}{6} $n(n + 1)(2n + 1)

$\displaystyle \sum\limits_{r = 1}^n {r^3} = \frac{1}{4} $n^{2}(n + 1)^{2}

$\displaystyle \sum\limits_{r = 1}^n {r\left( {r + 1} \right) } = \frac{1}{3}$ n(n + 1)(n + 2)

I can't figure out how to get from the right-hand side of the standard result to the final answer of $\displaystyle \frac{n}{n + 1}$. I assume I'm missing something really simple about the relationship between $\displaystyle \sum{f(x)}$ and $\displaystyle \sum{\frac{1}{f(x)}}$, but I just can't see it. Any help would be greatly appreciated. Thanks very much.